solve for $\log_x ( \frac{5}{2} - \frac{1}{x} ) \gt \frac{5}{2} - \frac{1}{x} $

Question

$\log_x ( \dfrac{5}{2} - \dfrac{1}{x} ) \gt \dfrac{5}{2} - \dfrac{1}{x} $

I found the domain which is $x \gt \dfrac{2}{5} \;$ and $x \neq 1.$

I tried making two cases $x \gt 1\;$ and $x\lt 1\;$ and solving it. But that only gives me a lot of terms.

I was also thinking about making a graph.

Answer 1

We get $$\frac{2}{5}<x<0.54568953436119576369$$ or $$1<x<1.38188769407987911316$$